Integrand size = 38, antiderivative size = 97 \[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {\cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c f (c-c \sin (e+f x))^{3/2}}+\frac {a \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/c/f/(c-c*sin(f*x+e))^(3/2)+a*cos(f*x+e)* ln(1-sin(f*x+e))/c^2/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 2.83 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {\sec (e+f x) \sqrt {a (1+\sin (e+f x))} \left (2-i f x+2 \log \left (i-e^{i (e+f x)}\right )+\left (i f x-2 \log \left (i-e^{i (e+f x)}\right )\right ) \sin (e+f x)\right )}{c^2 f \sqrt {c-c \sin (e+f x)}} \]
(Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*(2 - I*f*x + 2*Log[I - E^(I*(e + f*x))] + (I*f*x - 2*Log[I - E^(I*(e + f*x))])*Sin[e + f*x]))/(c^2*f*Sqrt[c - c*Sin[e + f*x]])
Time = 0.78 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {3042, 3320, 3042, 3218, 3042, 3216, 3042, 3146, 16}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{(c-c \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (e+f x)^2 \sqrt {a \sin (e+f x)+a}}{(c-c \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3320 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{3/2}}dx}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{3/2}}dx}{a c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx}{c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx}{c}}{a c}\) |
| \(\Big \downarrow \) 3216 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a^2 \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a^2 \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{a c}\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {\frac {a^2 \cos (e+f x) \int \frac {1}{c-c \sin (e+f x)}d(-c \sin (e+f x))}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}}{a c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {a^2 \cos (e+f x) \log (c-c \sin (e+f x))}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}}{a c}\) |
((a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*(c - c*Sin[e + f*x])^(3/2)) + (a^2*Cos[e + f*x]*Log[c - c*Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]] *Sqrt[c - c*Sin[e + f*x]]))/(a*c)
3.1.7.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[a*c*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x ]]*Sqrt[c + d*Sin[e + f*x]])) Int[Cos[e + f*x]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0 ]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(2*n + 1))), x] - Simp[b*((2*m - 1)/(d*( 2*n + 1))) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b ^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] && !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(a^(p/ 2)*c^(p/2)) Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2]
Time = 0.25 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\frac {\left (\ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )-2 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )+2 \sin \left (f x +e \right )-\ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+2 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sec \left (f x +e \right )}{f \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{2}}\) | \(127\) |
1/f*(ln(2/(1+cos(f*x+e)))*sin(f*x+e)-2*ln(csc(f*x+e)-cot(f*x+e)-1)*sin(f*x +e)+2*sin(f*x+e)-ln(2/(1+cos(f*x+e)))+2*ln(csc(f*x+e)-cot(f*x+e)-1))*(a*(1 +sin(f*x+e)))^(1/2)/(-c*(sin(f*x+e)-1))^(1/2)/c^2*sec(f*x+e)
\[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right ) + a} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
integral(-sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*cos(f*x + e)^ 2/(3*c^3*cos(f*x + e)^2 - 4*c^3 - (c^3*cos(f*x + e)^2 - 4*c^3)*sin(f*x + e )), x)
\[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \cos ^{2}{\left (e + f x \right )}}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right ) + a} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=-\frac {{\left (2 \, \sqrt {c} \log \left ({\left | \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + \frac {\sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}\right )} \sqrt {a}}{c^{3} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
-(2*sqrt(c)*log(abs(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*sgn(cos(-1/4*pi + 1/2 *f*x + 1/2*e)) + sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))/sin(-1/4*pi + 1/2*f*x + 1/2*e)^2)*sqrt(a)/(c^3*f*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))
Timed out. \[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]